[1]} summarizes a number of properties that help to classify robustness metrics.
Since most of these properties are especially relevant for optimization, we summarize them here for all considered robustness definitions in Table  REF  for the following property definitions.
A metric is sound if \(r(\varphi , x, t) \ge 0\)  if and only if the signal \(x\)  satisfies the specification at time \(t\) .
A metric is weakly smooth if it is continuous everywhere and its gradient is continuous for all points.
A metric is said to converge if it guaranteed to reach a local maximum with gradient-based optimization.
A metric is shadow-lifting if the metric increases when making partial progress towards the task specification.
A metric is scale-invariant if \(\operatorname{max}(\alpha a_1\wedge \alpha a_2) = \alpha \; \operatorname{max}(a_1\wedge a_2)\)  for any \(\alpha \in \mathbb {R}\) .
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